Importancia del trabajo por rincones

Having found the the eigenfunctions of the collective, symmetry broken Hamil- tonian, the question arises what these states represent, and even if they are truly physical states. As mentioned before, the symmetry breaking field in the collective Hamiltonian (4.19) acts as an implicit gauge fix. It is not a priori clear whether or not this (non-physical) gauge fixing introduced any extra unphysical states in the spectrum. If we define the gauge volume of a certain state to be the collection of all possible states that are connected to it by a gauge transformation, then the question is whether the excited states predicted byH_collSB are part of the ground state gauge volume or not.

The ground state of the collective Hamiltonian is a canted antiferromagnet in terms of pseudospins, and we have seen that it corresponds to a superconducting state of Cooper pairs. The excitations labeled byxin the pseudospin picture must involve the superposition of collective excitations with wavenumbersk = 0 and

k=π. However as mentioned before, the gauge volume of this system is made up of global uniform rotations of all of the pseudospins on the entire lattice around the

z-axis. We can thus prove that the excitations labelled byxare not within the gauge volume of the groundstate wavefunction, by showing that the excited states cannot be written as only a global rotation of the groundstate. To do so we will consider one specific thin spectrum state (the state withx=0), and do a gauge transformation on it by rotating it over an angleθ. We will then compare the resulting state with all other thin spectrum states (labelled withx =X) of the un-rotated model, and

4.2. THE LOCAL PAIRING SUPERCONDUCTOR 73 show that the overlap between the states is smaller than one for all possible choices ofXandθ. The thin spectrum withx=0 does therefore not contain a states which is merely a rotated version of the statex =0, and thus the thin spectrum states do not coincide with the ground state’s gauge volume. Using the explicit formulas for the eigenfunctions ofH_collSB it is easy to check that indeed the overlap between the state withx =X and the state withx =0, rotated over an angleθ, is one if and only if bothX and θare zero (see figure 4.5). This shows that indeed the excited state cannot be written only as a global rotation of the groundstate, and thus that the excited state is not within the ground state’s gauge volume.

Figure 4.5: The overlap between the thin spectrum state|xand the rotated ground- stateRˆ(θ )|0, as a function of the angle of rotationθ, for different values ofx. To make this graph we used the valuesJ =10,B=h=1 andN =100. For higher values ofN the graph for eachxwill be scaled horizontally, but the height of the top remains unaffected. The plot is symmetric under mirroring in theθ =0 axis. The inset shows the maximum of each curve, plotted as a function ofx. The max- imum of a curve also corresponds to the maximum overlap that a certain excited thin spectrum state has with (a gauge transformation of) the ground state. The solid line in the inset is included to guide the eye only.

The excitations labeled byywere already identified as corresponding to a change in the average total number of Cooper pairs in the superconductor. They alter thez

74 PART III, CHAPTER 4. THE SUPERCONDUCTOR

state’s gauge volume. These excited states are gapped and could thus be used to define an appropriate two-level system for building a qubit out of this local-pairing superconductor. Using that qubit we can then study its decoherence due to the existence of the (dual) thin spectrum formed by thexstates.

4.2.4

Decoherence

We would now like to apply the results of the previous section to the description of quantum coherence. In analogy to the result for antiferromagnets [23], we expect the existence of the unobservable thin spectrum to give rise a maximum coherence timetspon∝Nh/kBT.

Let us define a qubit made of the eigenstates of the collective part of the local pairing superconductor. If temperature is sufficiently low (i.e.kBTJ,h) then we

can use the statesy =0 andy=1 as the computational states of such a qubit. These states correspond to states with a different number of Cooper pairs, and qubits of this type have been made experimentally in the form of Cooper-pair boxes [79, 83, 122]. In these Cooper-pair boxes a superconducting island can be brought into a superposition of havingN¯ andN¯ +1 Cooper-pairs present. Superpositions of this type can reach coherence times of up to 500 ns [84, 123].

Figure 4.6: The Cooper-pair box qubit or ’quantronium’ studied in Saclay [79, 82]. The actual Cooper-pair box is the small rectangular island in the centre which can hold a superposition ofNandN+1 Cooper pairs.

4.2. THE LOCAL PAIRING SUPERCONDUCTOR 75 In our local pairing description of the qubit, the initial state of the system must be a thermal mixture of thin spectrum states. After all, controlling these states experimentally is practically impossible [51, 63]. The initial state should then be brought into some superposition of the computational statesy = 0 and y = 1, so that it can be used in a quantum computation. Because we know all eigenstates and eigenvalues of the Hamiltonian exactly, we can then explicitly follow the time evolution of the superposition [25]. The complete process is thus described by

ρt<0 = 1 Z x e−βE(x,0)|x, 0 x, 0| ρt=0 = 1 2Z x e−βE(x,0)[|x, 0 + |x, 1] [x, 0| + x, 1|] ρt>0 = 1 2Z x e−βE(x,0)[|x, 0 x, 0| + |x, 1 x, 1| +e−hi(E(x,0)−E(x,1))t|x, 0 x, 1| +h.c. . (4.24) where Z is the partition function att < 0. The thin spectrum states labeled by

x cannot be observed or controlled experimentally, and they should therefore be traced out of the final density matrix [94, 95]. The remaining reduced density ma- trix then shows the coherence of only the superposition of y states. The disap- pearance of the off-diagonal matrix element of the reduced density matrix serves as a measure of the resulting coherence time, and it can easily be checked that this coherence time is given by

tspon= 2πh kBT ¯ σ 2. (4.25)

Hereσ¯ signifies, as before, the average number of Cooper pairs on the supercon- ducting island in the groundstate. This coherence time is the maximum coherence time of a superconducting island, which is limited by the existence of a thin spec- trum in the superconductor. Just as in the cases of crystals and antiferromagnets, the details of the model (e.g. Jorh) do not enter into the expression for the maxi- mum coherence time, which thus looks like a universal timescale [23, 25].

Filling in the values for the constantshandkBand takingσ¯ 106andT40

mK [83], we find a coherence time for the experimentally realized Cooper pair boxes of 500µs. In fact this is a rather conservative estimate, since the elec- tronic temperature of the Cooper-pair box is probably higher then the environmen- tal temperature of 40 mK. Even so though, the timescale set by the presence of the thin spectrum states which are associated with the spontaneous symmetry break- ing, is clearly much larger than the current experimentally seen limit to coherence of the Cooper-pair boxes. This present limit is due to environmental factors, which

76 PART III, CHAPTER 4. THE SUPERCONDUCTOR

induce decoherence in times of the order of microseconds. However, it is well pos- sible that the limit set by the thin states will come within the experimental reach in the near future, either because the isolation from external sources of decoherence will be developed further, or because the size of the Cooper-pair box itself is reduced even more.